Školjka

Let $z_0 < z_1 < z_2 < \cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geq 1$ such that $$ z_n < \frac{z_0 + z_1 + \cdots + z_n}{n} \leq z_{n+1} \text{.} $$ \begin{flushright}\emph{(Austria)}\end{flushright}